Hypothesis Testing > F-Test

**Contents:**

**See also:** F Statistic in ANOVA/Regression

## What is an F Test?

An “F Test” is a catch-all term for **any test that uses the F-distribution**. In most cases, when people talk about the F-Test, what they are actually talking about is The *F-Test to Compare Two Variances. *However, the f-statistic is used in a variety of tests including regression analysis, the Chow test and the Scheffe Test (a post-hoc ANOVA test).

## General Steps for an F Test

If you’re running an F Test, you should use Excel, SPSS, Minitab or some other kind of technology to run the test. Why? Calculating the F test by hand, including variances, is tedious and time-consuming. Therefore you’ll probably make some errors along the way.

If you’re running an F Test using technology (for example, an F Test two sample for variances in Excel), the only steps you really need to do are Step 1 and 4 (dealing with the null hypothesis). Technology will calculate Steps 2 and 3 for you.

- State the null hypothesis and the alternate hypothesis.
- Calculate the F value. The F Value is calculated using the formula F = (SSE
_{1}– SSE_{2}/ m) / SSE_{2}/ n-k, where SSE = residual sum of squares, m = number of restrictions and k = number of independent variables. - Find the F Statistic (the critical value for this test). The F statistic formula is:

**F Statistic = variance of the group means / mean of the within group variances.**You can find the F Statistic in the F-Table.

- Support or Reject the Null Hypothesis.

## F Test to Compare Two Variances

A **Statistical F Test** uses an F Statistic to compare two variances, s_{1} and s_{2}, by dividing them. The result is always a positive number (because variances are always positive). The equation for comparing two variances with the f-test is:

F = s^{2}_{1} / s^{2}_{2}

If the variances are equal, the ratio of the variances will equal 1. For example, if you had two data sets with a sample 1 (variance of 10) and a sample 2 (variance of 10), the ratio would be 10/10 = 1.

You **always **test that the population variances are equal when running an F Test. In other words, you always assume that the variances are equal to 1. Therefore, your null hypothesis will always be that *the variances are equal*.

## Assumptions

Several **assumptions** are made for the test. Your population **must be approximately normally distributed ** (i.e. fit the shape of a bell curve) in order to use the test. Plus, the samples must be independent events. In addition, you’ll want to bear in mind a few important points:

- The larger variance should always go in the numerator (the top number) to force the test into a right-tailed test. Right-tailed tests are easier to calculate.
- For two-tailed tests, divide alpha by 2 before finding the right critical value.
- If you are given standard deviations, they must be squared to get the variances.
- If your degrees of freedom aren’t listed in the F Table, use the larger critical value. This helps to avoid the possibility of Type I errors.

## F Test to compare two variances by hand: Steps

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**Warning**: F tests can get really tedious to calculate by hand, especially if you have to calculate the variances. You’re much better off using technology (like Excel — see below).

**These are the general steps to follow. Scroll down for a specific example.**

**Step 1**: *If you are given standard deviations, go to Step 2. If you are given variances to compare, go to Step 3.*

**Step 2:** *Square both standard deviations to get the variances.* For example, if σ_{1} = 9.6 and σ_{2} = 10.9, then the variances (s_{1} and s_{2}) would be 9.6^{2} = **92.16** and 10.9^{2} = **118.81**.

**Step 3:*** Take the largest variance, and divide it by the smallest variance to get the f-value.* For example, if your two variances were s_{1} = 2.5 and s_{2} = 9.4, divide 9.4 / 2.5 = **3.76**.

Why? Placing the largest variance on top will force the F-test into a right tailed test, which is much easier to calculate than a left-tailed test.

**Step 4:** Find your degrees of freedom. Degrees of freedom is your sample size minus 1. As you have two samples (variance 1 and variance 2), you’ll have two degrees of freedom: one for the numerator and one for the denominator.

**Step 5:*** Look at the f-value you calculated in Step 3 in the f-table. *Note that there are several tables, so you’ll need to locate the right table for your alpha level. Unsure how to read an f-table? Read What is an f-table?.

**Step 6:** Compare your calculated value (Step 3) with the table f-value in Step 5. If the f-table value is smaller than the calculated value, you can reject the null hypothesis.

*That’s it!*

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## Two Tailed F-Test

The difference between running a one or two tailed F test is that the alpha level needs to be halved for two tailed F tests. For example, instead of working at α = 0.05, you use α = 0.025; Instead of working at α = 0.01, you use α = 0.005.

With a two tailed F test, you just want to know if the variances are not equal to each other. In notation:

H_{a} = σ^{2}1 ≠ σ^{2} 2

**Example problem: **Conduct a two tailed F Test on the following samples:

Sample 1: Variance = 109.63, sample size = 41.

Sample 2: Variance = 65.99, sample size = 21.

Step 1: Write your hypothesis statements:

H_{o}: No difference in variances.

H_{a}: Difference in variances.

Step 2: Calculate your F critical value. Put the highest variance as the numerator and the lowest variance as the denominator:

F Statistic = variance 1/ variance 2 = 109.63 / 65.99 = 1.66

Step 3: Calculate the degrees of freedom:

The degrees of freedom in the table will be the sample size -1, so:

Sample 1 has 40 df (the numerator).

Sample 2 has 20 df (the denominator).

Step 4: Choose an alpha level. No alpha was stated in the question, so use 0.05 (the standard “go to” in statistics). This needs to be halved for the two-tailed test, so use 0.025.

Step 5: Find the critical F Value using the F Table. There are several tables, so make sure you look in the alpha = .025 table. Critical F (40,20) at alpha (0.025) = 2.287.

Step 6: Compare your calculated value (Step 2) to your table value (Step 5). If your calculated value is higher than the table value, you can reject the null hypothesis:

F calculated value: 1.66

F value from table: 2.287.

1.66 < 2 .287.

So we cannot reject the null hypothesis.

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## F-Test to Compare Two Variances in Excel

### F-test two sample for variances Excel 2013: Steps

Step 1: Click the “Data” tab and then click “Data Analysis.”

Step 2: Click “F test two sample for variances” and then click “OK.”

Step 3: Click the Variable 1 Range box and then type the location for your first set of data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box.

Step 4: Click the Variable 2 box and then type the location for your second set of data. For example, if you typed your data into cells B1 to B10, type “B1:B10” into that box.

Step 5: Click the “Labels” box if your data has column headers.

Step 6: Choose an alpha level. In most cases, an alpha level of 0.05 is usually fine.

Step 7: Select a location for your output. For example, click the “New Worksheet” radio button.

Step 8: Click “OK.”

Step 9: Read the results. If your f-value is higher than your F critical value, reject the null hypothesis as your two populations have unequal variances.

**Warning:** Excel has a small “quirk.” Make sure that variance 1 is higher than variance 2. If it isn’t switch your input data around (i.e. make input 1 “B” and input 2 “A”). Otherwise, Excel will calculate an incorrect f-value. This is because the variance is a ratio of variance 1/variance 2, and Excel can’t work out which set of data is set 1 and set 2 without you explicitly telling it.

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## References

Archdeacon, T. (1994). Correlation and Regression Analysis: A Historian’s Guide. Univ of Wisconsin Press.